Subgroups — #28318796

Order 13568.

870 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊#424	C₂	1	1	yes	maximal, characteristic
2	#20291177	C₂	1	1	yes	maximal, characteristic
3	#20290513	C₂	1	1	yes	maximal, characteristic, complemented
4	#20290513	C₂	1	1	yes	maximal, characteristic, complemented
5	#20290509	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₄⋊#6157	C₂	1	1	yes	maximal, characteristic, complemented
7	#20290504	C₂	1	1	yes	maximal, characteristic, complemented
8	#20290504	C₂	1	1	yes	maximal, characteristic, complemented
9	#20290502	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×#6157	C₂	1	1	yes	maximal, characteristic, complemented
11	#20290458	C₂	1	1	yes	maximal, characteristic
12	#20290458	C₂	1	1	yes	maximal, characteristic
13	#20290454	C₂	1	1	yes	maximal, characteristic
14	#20290450	C₂	1	1	yes	maximal, characteristic
15	#433×D₅₃	C₂	1	1	yes	maximal, characteristic
16	C₂×D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
18	#16811630	C₄	1	2	yes	complemented
19	#16811630	C₄	1	2	yes	complemented
20	#16811629	C₄	1	2	yes	complemented
21	#16811629	C₄	1	2	yes	complemented
22	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
23	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
24	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
25	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
26	C₂×#13133271	C₂²	1	1	yes	characteristic
27	C₂×#13133271	C₂²	1	1	yes	characteristic
28	C₂×#13133271	C₂²	1	1	yes	characteristic
29	C₂×#13133271	C₂²	1	1	yes	characteristic
30	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
31	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
32	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
33	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
34	#16811573	C₄	1	2	yes	—
35	#16811573	C₄	1	2	yes	—
36	#16811572	C₄	1	2	yes	—
37	#16811572	C₄	1	2	yes	—
38	C₄×C₄⋊#6157	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊#6157	C₂²	1	1	yes	characteristic
40	C₄×C₂²⋊#6157	C₂²	1	1	yes	characteristic
41	C₄×C₂²⋊#6157	C₂²	1	1	yes	characteristic
42	#16811559	C₂²	1	1	yes	characteristic
43	C₄²×#6157	C₂²	1	1	yes	characteristic
44	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
45	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
46	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
47	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
48	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
52	#117×D₅₃	C₂²	1	1	yes	characteristic
53	#16810685	C₄	1	1	yes	characteristic, complemented
54	#16810684	C₄	1	1	yes	characteristic, complemented
55	#16810683	C₄	1	1	yes	characteristic, complemented
56	#16810683	C₄	1	1	yes	characteristic, complemented
57	#16810682	C₄	1	1	yes	characteristic, complemented
58	#16810681	C₄	1	1	yes	characteristic, complemented
59	#16810681	C₄	1	1	yes	characteristic, complemented
60	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
61	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
62	C₂²⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
63	C₂²⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
64	#97×D₅₃	C₂²	1	1	yes	characteristic
65	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
66	C₅₃×#433	C₄	1	1	yes	characteristic, complemented
67	C₂×#11719936	C₂×C₄	1	2	yes	—
68	D₅₃⋊M₁₆	—	4	1	no	—
69	D₅₃⋊M₁₆	—	4	1	no	—
70	C₂×#11719940	C₂×C₄	1	2	yes	—
71	C₂×D₅₃⋊C₈	—	4	1	no	—
72	C₂×D₅₃⋊C₈	—	4	1	no	—
73	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
74	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂²×#88324	C₂×C₄	1	2	yes	—
79	C₂×C₄⋊#6157	C₂×C₄	1	2	yes	—
80	C₂×C₄⋊#6157	C₂×C₄	1	2	yes	—
81	C₂×C₂²⋊#6157	C₂×C₄	1	2	yes	—
82	C₂×C₂²⋊#6157	C₂×C₄	1	2	yes	—
83	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
84	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
85	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
86	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
87	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
88	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
89	#13133284	C₂×C₄	1	2	yes	—
90	#13133284	C₂×C₄	1	2	yes	—
91	#13133284	C₂×C₄	1	2	yes	—
92	#13133284	C₂×C₄	1	2	yes	—
93	C₄⋊#88324	C₂×C₄	1	2	yes	—
94	C₄⋊#88324	C₂×C₄	1	2	yes	—
95	C₄⋊#88324	C₂×C₄	1	2	yes	—
96	C₄⋊#88324	C₂×C₄	1	2	yes	—
97	#13133271	—	4	1	no	—
98	#13133271	—	4	1	no	—
99	#13133271	—	4	1	no	—
100	#13133271	—	4	1	no	—
101	C₈×#6157	—	4	1	no	—
102	C₈×#6157	—	4	1	no	—
103	C₈×#6157	—	4	1	no	—
104	C₈×#6157	—	4	1	no	—
105	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
106	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
107	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
108	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
109	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
110	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
111	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
112	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
113	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
114	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
115	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
116	#13133176	C₂×C₄	1	1	yes	characteristic
117	#13133175	C₂×C₄	1	1	yes	characteristic
118	#13133175	C₂×C₄	1	1	yes	characteristic
119	#13133174	C₂×C₄	1	1	yes	characteristic
120	#13133174	C₂×C₄	1	1	yes	characteristic
121	#13133173	C₂×C₄	1	1	yes	characteristic
122	#13133173	C₂×C₄	1	1	yes	characteristic
123	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
124	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
125	C₂²⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
126	C₂²⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
127	C₄²×D₅₃	C₂³	1	1	yes	characteristic
128	C₄²×D₅₃	C₂³	1	1	yes	characteristic
129	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
130	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
131	#13133146	C₂×C₄	1	1	yes	characteristic
132	#13133146	C₂×C₄	1	1	yes	characteristic
133	#13133146	C₂×C₄	1	1	yes	characteristic
134	#13133146	C₂×C₄	1	1	yes	characteristic
135	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
136	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
137	#13133129	C₂×C₄	1	1	yes	characteristic
138	#13133129	C₂×C₄	1	1	yes	characteristic
139	#13133129	C₂×C₄	1	1	yes	characteristic
140	#13133129	C₂×C₄	1	1	yes	characteristic
141	#13133128	C₂×C₄	1	1	yes	characteristic
142	#13133127	C₂×C₄	1	1	yes	characteristic
143	#13133127	C₂×C₄	1	1	yes	characteristic
144	C₄×#88322	C₂×C₄	1	1	yes	characteristic
145	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
146	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
147	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
148	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
149	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
150	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
151	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
152	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
153	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
154	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×#6157	C₂²×C₄	1	2	yes	—
156	C₂²×#6157	C₂²×C₄	1	2	yes	—
157	C₂²×#6157	C₂²×C₄	1	2	yes	—
158	C₂²×#6157	C₂²×C₄	1	2	yes	—
159	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₂×C₄×D₅₃	—	2	1	no	—
162	C₂×C₄×D₅₃	—	2	1	no	—
163	C₂×C₄×D₅₃	—	2	1	no	—
164	C₂×C₄×D₅₃	—	2	1	no	—
165	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
166	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
167	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
172	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
173	C₂²⋊#6157	—	2	4	no	—
174	C₂²⋊#6157	—	4	2	no	—
175	#11719940	—	4	2	no	—
176	C₂×#88324	—	4	2	no	—
177	C₂×#88324	C₂²×C₄	1	2	yes	—
178	C₂×#88324	C₂²×C₄	1	2	yes	—
179	C₂×#88324	C₂²×C₄	1	2	yes	—
180	C₂×#88324	C₂²×C₄	1	2	yes	—
181	C₄⋊#6157	—	2	4	no	—
182	C₄⋊#6157	—	4	2	no	—
183	C₄×#6157	—	2	4	no	—
184	C₄×#6157	—	2	1	no	—
185	C₄×#6157	—	2	1	no	—
186	C₄×#6157	—	2	1	no	—
187	C₄×#6157	—	2	1	no	—
188	C₄×#6157	—	2	1	no	—
189	C₄×#6157	—	2	1	no	—
190	C₄×#6157	—	2	1	no	—
191	C₄×#6157	—	2	1	no	—
192	C₄×#6157	—	4	2	no	—
193	#11719936	—	4	2	no	—
194	D₅₃⋊C₈	—	2	1	no	—
195	D₅₃⋊C₈	—	2	1	no	—
196	D₅₃⋊C₈	—	2	1	no	—
197	D₅₃⋊C₈	—	2	1	no	—
198	D₅₃⋊C₈	—	2	1	no	—
199	D₅₃⋊C₈	—	2	1	no	—
200	D₅₃⋊C₈	—	2	1	no	—
201	D₅₃⋊C₈	—	2	1	no	—
202	D₅₃⋊C₈	—	4	2	no	—
203	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
204	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
205	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
206	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
207	C₄⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
208	C₄⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
209	C₂²⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
210	C₂²⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
211	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
212	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
213	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
214	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
215	#11719921	C₄²	1	1	yes	characteristic
216	#11719921	C₄²	1	1	yes	characteristic
217	#11719921	C₄²	1	1	yes	characteristic
218	#11719921	C₄²	1	1	yes	characteristic
219	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
220	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
221	#11719919	C₄²	1	1	yes	characteristic
222	#11719919	C₄²	1	1	yes	characteristic
223	#11719919	C₄²	1	1	yes	characteristic
224	#11719919	C₄²	1	1	yes	characteristic
225	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
226	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
227	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
228	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
229	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
230	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
231	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
232	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
233	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
234	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
235	C₈×D₅₃	—	2	1	no	—
236	C₈×D₅₃	—	2	1	no	—
237	C₈×D₅₃	—	2	1	no	—
238	C₈×D₅₃	—	2	1	no	—
239	C₈×D₅₃	—	2	1	no	—
240	C₈×D₅₃	—	2	1	no	—
241	C₈×D₅₃	—	2	1	no	—
242	C₈×D₅₃	—	2	1	no	—
243	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
244	C₂²×D₅₃	—	2	1	no	—
245	C₂²×D₅₃	—	2	1	no	—
246	C₂²×D₅₃	—	2	2	no	—
247	C₂²×D₅₃	—	2	2	no	—
248	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
249	C₂²×D₅₃	C₂³×C₄	1	1	yes	characteristic
250	C₂×#6157	—	2	2	no	—
251	C₂×#6157	—	2	2	no	—
252	C₂×#6157	—	2	2	no	—
253	C₂×#6157	—	2	2	no	—
254	C₂×#6157	—	2	2	no	—
255	C₂×#6157	—	2	2	no	—
256	C₂×#6157	—	2	2	no	—
257	C₂×#6157	—	2	2	no	—
258	C₂×#6157	—	2	2	no	—
259	C₂×#6157	—	2	2	no	—
260	C₂×#6157	—	2	2	no	—
261	C₂×#6157	#131	1	4	yes	—
262	C₂×C₄×C₅₃	—	2	1	no	—
263	C₂×C₄×C₅₃	—	2	1	no	—
264	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
265	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
266	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
267	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
268	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
269	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
270	C₂×Dic₅₃	—	2	1	no	—
271	C₂×Dic₅₃	—	2	1	no	—
272	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
273	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
274	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
275	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
276	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
277	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
278	C₄×D₅₃	—	2	1	no	—
279	C₄×D₅₃	—	2	1	no	—
280	C₄×D₅₃	—	2	2	no	—
281	C₄×D₅₃	—	2	2	no	—
282	C₄×D₅₃	—	2	2	no	—
283	C₄×D₅₃	—	2	2	no	—
284	C₄×D₅₃	—	2	1	no	—
285	C₄×D₅₃	—	2	1	no	—
286	C₄×D₅₃	—	2	1	no	—
287	C₄×D₅₃	—	2	1	no	—
288	C₄×D₅₃	H(5,𝔽₂)	1	1	yes	characteristic
289	C₄×D₅₃	H(5,𝔽₂)	1	1	yes	characteristic
290	C₄×D₅₃	2₋¹⁺⁴	1	1	yes	characteristic
291	C₄×D₅₃	2₋¹⁺⁴	1	1	yes	characteristic
292	#88324	—	2	2	no	—
293	#88324	—	2	2	no	—
294	#88324	—	2	2	no	—
295	#88324	—	2	2	no	—
296	C₈×C₅₃	—	2	1	no	—
297	C₈×C₅₃	—	2	1	no	—
298	C₈×C₅₃	—	2	1	no	—
299	C₈×C₅₃	—	2	1	no	—
300	#88322	—	2	1	no	—
301	#88322	—	2	1	no	—
302	#88322	—	2	1	no	—
303	#88322	—	2	1	no	—
304	#20553	—	53	1	no	maximal, Sylow
305	C₂²×C₅₃	—	2	1	no	—
306	C₂²×C₅₃	—	2	1	no	—
307	C₂²×C₅₃	—	2	1	no	—
308	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
309	C₂×D₅₃	—	2	1	no	—
310	C₂×D₅₃	—	2	1	no	—
311	C₂×D₅₃	—	2	1	no	—
312	C₂×D₅₃	—	4	2	no	—
313	C₂×D₅₃	—	4	2	no	—
314	C₂×D₅₃	#520	1	1	yes	characteristic
315	C₂×D₅₃	#520	1	1	yes	characteristic
316	C₂×D₅₃	C₂×#131	1	2	yes	—
317	C₂×D₅₃	#568	1	2	yes	—
318	#6157	—	2	4	no	—
319	#6157	—	4	2	no	—
320	#6157	—	4	2	no	—
321	#6157	—	4	2	no	—
322	C₄×C₅₃	—	2	1	no	—
323	C₄×C₅₃	—	2	1	no	—
324	C₄×C₅₃	—	2	1	no	—
325	C₄×C₅₃	—	2	1	no	—
326	C₄×C₅₃	—	2	1	no	—
327	C₄×C₅₃	#520	1	1	yes	characteristic
328	C₄×C₅₃	#520	1	1	yes	characteristic
329	Dic₅₃	—	2	1	no	—
330	Dic₅₃	—	2	1	no	—
331	Dic₅₃	—	2	1	no	—
332	Dic₅₃	—	2	1	no	—
333	Dic₅₃	—	2	1	no	—
334	Dic₅₃	#518	1	1	yes	characteristic
335	Dic₅₃	#519	1	1	yes	characteristic
336	C₂×#433	—	53	1	no	—
337	C₂×#424	—	53	1	no	—
338	#2298	—	53	1	no	—
339	#1771	—	53	1	no	—
340	#1771	—	53	1	no	—
341	#1770	—	53	1	no	—
342	C₈×C₄⋊C₄	—	53	1	no	—
343	#1768	—	53	1	no	—
344	#1768	—	53	1	no	—
345	#1767	—	53	1	no	—
346	C₄×C₄⋊C₈	—	53	1	no	—
347	#1753	—	53	1	no	—
348	#1753	—	53	1	no	—
349	#1752	—	53	1	no	—
350	C₈×C₂²⋊C₄	—	53	1	no	—
351	C₂×C₅₃	—	4	1	no	—
352	C₂×C₅₃	#2274	1	1	yes	characteristic
353	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
354	C₂×C₅₃	#2875	1	1	yes	characteristic
355	D₅₃	—	4	1	no	—
356	D₅₃	#2992	1	2	yes	—
357	D₅₃	#2993	1	2	yes	—
358	C₂²×M₁₆	—	53	1	no	—
359	C₂³×C₈	—	53	1	no	—
360	C₂×#117	—	53	1	no	—
361	#433	—	53	1	no	—
362	#433	—	53	1	no	—
363	#433	—	53	1	no	—
364	#433	—	53	1	no	—
365	#433	—	53	1	no	—
366	#433	—	53	1	no	—
367	#433	—	53	1	no	—
368	#433	#6157	1	1	yes	characteristic, complemented
369	#424	—	53	2	no	—
370	#424	—	53	2	no	—
371	#424	—	53	2	no	—
372	#424	—	53	2	no	—
373	C₂×C₄⋊C₈	—	53	1	no	—
374	C₂×C₄⋊C₈	—	53	1	no	—
375	C₂×C₄⋊C₈	—	53	1	no	—
376	C₂×C₄⋊C₈	—	53	1	no	—
377	C₂×C₄⋊C₈	—	53	1	no	—
378	C₂×C₄⋊C₈	—	53	1	no	—
379	C₂×C₂²⋊C₈	—	53	1	no	—
380	C₂×C₂²⋊C₈	—	53	1	no	—
381	C₂×#97	—	53	1	no	—
382	C₂×#97	—	53	1	no	—
383	C₂×#97	—	53	1	no	—
384	C₂×#97	—	53	1	no	—
385	C₂×#97	—	53	1	no	—
386	C₂×C₄×C₈	—	53	1	no	—
387	C₂×C₄×C₈	—	53	1	no	—
388	C₂×C₄×C₈	—	53	1	no	—
389	C₂×C₄×C₈	—	53	1	no	—
390	C₂×C₄×C₈	—	53	1	no	—
391	#388	—	53	2	no	—
392	#388	—	53	2	no	—
393	#387	—	53	2	no	—
394	#387	—	53	2	no	—
395	C₄×C₄⋊C₄	—	53	1	no	—
396	C₄×C₄⋊C₄	—	53	1	no	—
397	C₄×C₂²⋊C₄	—	53	1	no	—
398	C₄×C₂²⋊C₄	—	53	1	no	—
399	#376	—	53	1	no	—
400	C₄³	—	53	1	no	—
401	#336	—	53	1	no	—
402	#336	—	53	1	no	—
403	#336	—	53	1	no	—
404	#336	—	53	1	no	—
405	#336	—	53	1	no	—
406	#336	—	53	1	no	—
407	#336	—	53	1	no	—
408	#336	—	53	1	no	—
409	C₅₃	#20553	1	1	yes	characteristic, complemented, Sylow
410	C₂³×C₄	—	53	1	no	—
411	C₂×M₁₆	—	53	2	no	—
412	C₂×M₁₆	—	53	2	no	—
413	C₂×M₁₆	—	212	1	no	—
414	C₂×M₁₆	—	212	1	no	—
415	C₂²×C₈	—	53	1	no	—
416	C₂²×C₈	—	53	1	no	—
417	C₂²×C₈	—	53	1	no	—
418	C₂²×C₈	—	53	1	no	—
419	C₂²×C₈	—	53	2	no	—
420	C₂²×C₈	—	53	2	no	—
421	C₂²×C₈	—	53	1	no	—
422	C₂²×C₈	—	53	1	no	—
423	C₂²×C₈	—	53	1	no	—
424	C₂²×C₈	—	53	1	no	—
425	C₂²×C₈	—	212	1	no	—
426	C₂²×C₈	—	212	1	no	—
427	#117	—	53	1	no	—
428	#117	—	53	1	no	—
429	#117	—	53	1	no	—
430	#117	—	53	1	no	—
431	#117	—	53	1	no	—
432	#117	—	53	1	no	—
433	#117	—	53	1	no	—
434	#117	C₂×#6157	1	1	yes	characteristic
435	C₂×C₄⋊C₄	—	53	1	no	—
436	C₂×C₄⋊C₄	—	53	1	no	—
437	C₂×C₄⋊C₄	—	53	2	no	—
438	C₂×C₄⋊C₄	—	53	2	no	—
439	C₂×C₂²⋊C₄	—	53	1	no	—
440	C₂×C₂²⋊C₄	—	53	1	no	—
441	C₂×C₂²⋊C₄	—	53	2	no	—
442	C₂×C₂²⋊C₄	—	53	2	no	—
443	C₂×C₄²	—	53	1	no	—
444	C₂×C₄²	—	53	1	no	—
445	C₂×C₄²	—	53	1	no	—
446	C₂×C₄²	—	53	1	no	—
447	C₂×C₄²	—	53	1	no	—
448	C₂×C₄²	—	53	1	no	—
449	C₂×C₄²	—	53	2	no	—
450	C₂×C₄²	—	53	2	no	—
451	C₄⋊C₈	—	53	2	no	—
452	C₄⋊C₈	—	53	2	no	—
453	C₄⋊C₈	—	53	2	no	—
454	C₄⋊C₈	—	53	2	no	—
455	C₄⋊C₈	—	53	2	no	—
456	C₄⋊C₈	—	53	2	no	—
457	C₄⋊C₈	—	53	2	no	—
458	C₄⋊C₈	—	53	2	no	—
459	C₄⋊C₈	—	53	1	no	—
460	C₄⋊C₈	—	53	1	no	—
461	C₄⋊C₈	—	53	1	no	—
462	C₄⋊C₈	—	53	1	no	—
463	C₄⋊C₈	—	53	1	no	—
464	C₄⋊C₈	—	53	1	no	—
465	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
466	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
467	C₂²⋊C₈	—	53	1	no	—
468	C₂²⋊C₈	—	53	1	no	—
469	C₂²⋊C₈	—	53	1	no	—
470	C₂²⋊C₈	—	53	1	no	—
471	C₂²⋊C₈	—	53	1	no	—
472	C₂²⋊C₈	—	53	1	no	—
473	C₂²⋊C₈	C₂×#6157	1	1	yes	characteristic
474	C₂²⋊C₈	C₂×#6157	1	1	yes	characteristic
475	#97	—	53	1	no	—
476	#97	—	53	1	no	—
477	#97	—	53	1	no	—
478	#97	—	212	1	no	—
479	#97	—	212	1	no	—
480	#97	—	212	1	no	—
481	#97	—	212	1	no	—
482	#97	C₂×#6157	1	1	yes	characteristic
483	C₄×C₈	—	53	1	no	—
484	C₄×C₈	—	53	1	no	—
485	C₄×C₈	—	53	1	no	—
486	C₄×C₈	—	212	1	no	—
487	C₄×C₈	—	212	1	no	—
488	C₄×C₈	—	212	1	no	—
489	C₄×C₈	—	212	1	no	—
490	C₄×C₈	C₂×#6157	1	1	yes	characteristic
491	#95	—	53	2	no	—
492	#95	—	53	2	no	—
493	#95	—	53	2	no	—
494	#95	—	53	2	no	—
495	#95	—	53	2	no	—
496	#95	—	53	2	no	—
497	C₂⁴	—	53	1	no	—
498	C₂²×C₄	—	53	1	no	—
499	C₂²×C₄	—	53	2	no	—
500	C₂²×C₄	—	53	1	no	—
501	C₂²×C₄	—	53	1	no	—
502	C₂²×C₄	—	53	1	no	—
503	C₂²×C₄	—	53	1	no	—
504	C₂²×C₄	—	53	1	no	—
505	C₂²×C₄	—	53	1	no	—
506	C₂²×C₄	—	53	1	no	—
507	C₂²×C₄	—	53	1	no	—
508	C₂²×C₄	—	53	2	no	—
509	C₂²×C₄	—	53	2	no	—
510	C₂²×C₄	—	53	2	no	—
511	C₂²×C₄	—	106	1	no	—
512	C₂²×C₄	—	106	1	no	—
513	C₂²×C₄	—	106	1	no	—
514	C₂²×C₄	—	106	1	no	—
515	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
516	M₁₆	—	212	2	no	—
517	M₁₆	—	212	2	no	—
518	C₂×C₈	—	53	2	no	—
519	C₂×C₈	—	53	2	no	—
520	C₂×C₈	—	53	2	no	—
521	C₂×C₈	—	53	2	no	—
522	C₂×C₈	—	53	1	no	—
523	C₂×C₈	—	53	1	no	—
524	C₂×C₈	—	53	1	no	—
525	C₂×C₈	—	53	1	no	—
526	C₂×C₈	—	106	1	no	—
527	C₂×C₈	—	106	1	no	—
528	C₂×C₈	—	106	1	no	—
529	C₂×C₈	—	106	1	no	—
530	C₂×C₈	—	106	1	no	—
531	C₂×C₈	—	106	1	no	—
532	C₂×C₈	—	106	1	no	—
533	C₂×C₈	—	106	1	no	—
534	C₂×C₈	—	106	1	no	—
535	C₂×C₈	—	106	1	no	—
536	C₂×C₈	—	106	1	no	—
537	C₂×C₈	—	106	1	no	—
538	C₂×C₈	—	106	1	no	—
539	C₂×C₈	—	106	1	no	—
540	C₂×C₈	—	106	1	no	—
541	C₂×C₈	—	106	1	no	—
542	C₂×C₈	—	212	2	no	—
543	C₂×C₈	—	212	2	no	—
544	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
545	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
546	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
547	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
548	C₄⋊C₄	—	53	1	no	—
549	C₄⋊C₄	—	53	1	no	—
550	C₄⋊C₄	—	53	1	no	—
551	C₄⋊C₄	—	53	1	no	—
552	C₄⋊C₄	—	53	1	no	—
553	C₄⋊C₄	—	53	1	no	—
554	C₄⋊C₄	—	106	4	no	—
555	C₄⋊C₄	—	212	2	no	—
556	C₄⋊C₄	C₄×#6157	1	1	yes	characteristic
557	C₄⋊C₄	C₄×#6157	1	1	yes	characteristic
558	C₂²⋊C₄	—	53	1	no	—
559	C₂²⋊C₄	—	53	1	no	—
560	C₂²⋊C₄	—	53	1	no	—
561	C₂²⋊C₄	—	53	1	no	—
562	C₂²⋊C₄	—	53	1	no	—
563	C₂²⋊C₄	—	53	1	no	—
564	C₂²⋊C₄	—	106	4	no	—
565	C₂²⋊C₄	—	212	2	no	—
566	C₂²⋊C₄	C₄×#6157	1	1	yes	characteristic
567	C₂²⋊C₄	C₄×#6157	1	1	yes	characteristic
568	C₄²	—	53	1	no	—
569	C₄²	—	53	1	no	—
570	C₄²	—	53	1	no	—
571	C₄²	—	53	1	no	—
572	C₄²	—	53	1	no	—
573	C₄²	—	53	1	no	—
574	C₄²	—	106	1	no	—
575	C₄²	—	106	1	no	—
576	C₄²	—	106	1	no	—
577	C₄²	—	106	1	no	—
578	C₄²	—	106	1	no	—
579	C₄²	—	106	1	no	—
580	C₄²	—	106	1	no	—
581	C₄²	—	106	1	no	—
582	C₄²	—	106	4	no	—
583	C₄²	—	212	2	no	—
584	C₄²	C₂²×#6157	1	1	yes	characteristic
585	C₄²	C₂²×#6157	1	1	yes	characteristic
586	C₂³	—	53	1	no	—
587	C₂³	—	53	1	no	—
588	C₂³	—	106	2	no	—
589	C₂³	—	106	2	no	—
590	C₂³	—	106	1	no	—
591	C₂³	—	106	1	no	—
592	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	53	1	no	—
596	C₂×C₄	—	53	1	no	—
597	C₂×C₄	—	53	1	no	—
598	C₂×C₄	—	53	1	no	—
599	C₂×C₄	—	53	1	no	—
600	C₂×C₄	—	53	4	no	—
601	C₂×C₄	—	53	1	no	—
602	C₂×C₄	—	53	1	no	—
603	C₂×C₄	—	53	1	no	—
604	C₂×C₄	—	53	1	no	—
605	C₂×C₄	—	53	1	no	—
606	C₂×C₄	—	106	1	no	—
607	C₂×C₄	—	106	1	no	—
608	C₂×C₄	—	106	2	no	—
609	C₂×C₄	—	106	2	no	—
610	C₂×C₄	—	106	2	no	—
611	C₂×C₄	—	106	2	no	—
612	C₂×C₄	—	106	1	no	—
613	C₂×C₄	—	106	1	no	—
614	C₂×C₄	—	106	1	no	—
615	C₂×C₄	—	106	1	no	—
616	C₂×C₄	—	106	2	no	—
617	C₂×C₄	—	106	2	no	—
618	C₂×C₄	—	106	1	no	—
619	C₂×C₄	—	106	1	no	—
620	C₂×C₄	—	106	2	no	—
621	C₂×C₄	—	106	2	no	—
622	C₂×C₄	—	106	2	no	—
623	C₂×C₄	—	106	2	no	—
624	C₂×C₄	—	106	2	no	—
625	C₂×C₄	—	106	2	no	—
626	C₂×C₄	—	106	2	no	—
627	C₂×C₄	—	106	2	no	—
628	C₂×C₄	—	106	2	no	—
629	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
630	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
631	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
632	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
633	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
634	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
635	C₈	—	2	1	no	—
636	C₈	—	2	1	no	—
637	C₈	—	2	1	no	—
638	C₈	—	2	1	no	—
639	C₈	—	106	2	no	—
640	C₈	—	106	2	no	—
641	C₈	—	106	2	no	—
642	C₈	—	106	2	no	—
643	C₈	—	106	1	no	—
644	C₈	—	106	1	no	—
645	C₈	—	106	1	no	—
646	C₈	—	106	1	no	—
647	C₂²	—	2	1	no	—
648	C₂²	—	2	1	no	—
649	C₂²	—	2	1	no	—
650	C₂²	—	53	1	no	—
651	C₂²	—	53	1	no	—
652	C₂²	—	53	2	no	—
653	C₂²	—	53	2	no	—
654	C₂²	—	106	1	no	—
655	C₂²	—	106	1	no	—
656	C₂²	—	106	1	no	—
657	C₂²	—	212	2	no	—
658	C₂²	—	212	2	no	—
659	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
660	C₄	—	2	1	no	—
661	C₄	—	2	1	no	—
662	C₄	—	2	1	no	—
663	C₄	—	2	1	no	—
664	C₄	—	2	1	no	—
665	C₄	—	53	1	no	—
666	C₄	—	53	1	no	—
667	C₄	—	106	4	no	—
668	C₄	—	106	1	no	—
669	C₄	—	106	1	no	—
670	C₄	—	106	1	no	—
671	C₄	—	106	1	no	—
672	C₄	—	106	1	no	—
673	C₄	—	212	2	no	—
674	C₄	—	212	2	no	—
675	C₄	—	212	2	no	—
676	C₄	#16811692	1	1	yes	characteristic
677	C₄	#16811692	1	1	yes	characteristic
678	C₂	—	4	1	no	—
679	C₂	—	53	2	no	—
680	C₂	—	53	2	no	—
681	C₂	—	212	1	no	—
682	C₂	#20291152	1	1	yes	characteristic
683	C₂	#131×#6157	1	1	yes	characteristic
684	C₂	#20291285	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#433	C₄
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#433	#6157
hypocenter	C₅₃	#20553
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #20290513	C₂	semidirect
4. #20290513	C₂	semidirect
5. #20290509	C₂	semidirect
6. C₈×C₄⋊#6157	C₂	semidirect
7. #20290504	C₂	semidirect
8. #20290504	C₂	semidirect
9. #20290502	C₂	semidirect
10. C₄⋊C₈×#6157	C₂	semidirect
18. #16811630	C₄	semidirect
19. #16811630	C₄	semidirect
20. #16811629	C₄	semidirect
21. #16811629	C₄	semidirect
53. #16810685	C₄	semidirect
54. #16810684	C₄	semidirect
55. #16810683	C₄	semidirect
56. #16810683	C₄	semidirect
57. #16810682	C₄	semidirect
58. #16810681	C₄	semidirect
59. #16810681	C₄	semidirect
66. C₅₃×#433	C₄	semidirect
368. #433	#6157	semidirect
409. C₅₃	#20553	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
D₅₃⋊#424	2
#20291177	2
#20290513	2
#20290509	2
C₈×C₄⋊#6157	2
#20290504	2
#20290502	2
C₄⋊C₈×#6157	2
#20290458	2
#20290454	2
#20290450	2
#433×D₅₃	2
#20553	53